Participants in this research study have been diagnosed with cirrhosis and increased pressure in the veins supplying blood to the liver (portal hypertension). Due to this increased back pressure in the veins bringing blood into the liver, fluid has accumulated in the abdominal cavity (ascites) which is not responding to treatment with water pills (diuretics) and is requiring frequent drainage of this fluid. This has also caused large varicose veins to develop along the lining of the esophagus, stomach and/or intestine. The purpose of this study is to determine if a relatively new investigational (experimental) procedure called transjugular intrahepatic portosystemic shunt (TIPS) can treat the severe fluid build up in the abdomen more effectively than the standard drainage procedures. In other patients with advanced liver disease who have had internal bleeding from veins in the esophagus under abnormally high-pressure (esophageal varices), the TIPS procedure decreases further bleeding by diverting blood flow from these veins. The purpose of this study is to determine if a similar diversion of blood from the veins with abnormally high pressure flowing into the liver with the TIPS will effectively treat abdominal fluid accumulation. The duration of study participation will be one year after beginning the study with subsequent follow-up until termination of the study, which will be up to 3 years after entry.